reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;
reserve X, Y, Z for ManySortedSet of I;

theorem :: CQC_THE1:14
  X is finite-yielding & X c= [|Y,Z|] implies ex A st A is
  finite-yielding & A c= Y & X c= [|A,Z|]
proof
  assume that
A1: X is finite-yielding and
A2: X c= [|Y,Z|];
  defpred P[object,object] means
    ex D being set st D = $2 & D is finite & D c= Y.$1 & X.$1 c= [:D,Z.$1:];
A3: for i being object st i in I ex j be object st P[i,j]
  proof
    let i be object;
    assume
A4: i in I;
    then X.i c= [|Y,Z|].i by A2;
    then
A5: X.i c= [:Y.i, Z.i:] by A4,PBOOLE:def 16;
    X.i is finite by A1;
    then consider A9 be set such that
A6: A9 is finite & A9 c= Y.i & X.i c= [:A9,Z.i:] by A5,FINSET_1:14;
    take A9;
    thus thesis by A6;
  end;
  consider A such that
A7: for i being object st i in I holds P[i,A.i] from PBOOLE:sch 3(A3);
  take A;
  thus A is finite-yielding
  proof
    let i be object;
    assume i in I;
     then P[i,A.i] by A7;
    hence thesis;
  end;
  thus A c= Y
  proof
    let i be object;
    assume i in I;
     then P[i,A.i] by A7;
    hence thesis;
  end;
  thus X c= [|A,Z|]
  proof
    let i be object;
    assume
A8: i in I;
     then P[i,A.i] by A7;
    then X.i c= [:A.i,Z.i:];
    hence thesis by A8,PBOOLE:def 16;
  end;
end;
